Integrand size = 18, antiderivative size = 131 \[ \int \frac {\left (a+\frac {b}{x}\right )^n x}{c+d x} \, dx=\frac {\left (a+\frac {b}{x}\right )^{1+n} x}{a d}+\frac {c^2 \left (a+\frac {b}{x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d^2 (a c-b d) (1+n)}-\frac {(a c-b d n) \left (a+\frac {b}{x}\right )^{1+n} \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b}{a x}\right )}{a^2 d^2 (1+n)} \]
(a+b/x)^(1+n)*x/a/d+c^2*(a+b/x)^(1+n)*hypergeom([1, 1+n],[2+n],c*(a+b/x)/( a*c-b*d))/d^2/(a*c-b*d)/(1+n)-(-b*d*n+a*c)*(a+b/x)^(1+n)*hypergeom([1, 1+n ],[2+n],1+b/a/x)/a^2/d^2/(1+n)
Time = 0.24 (sec) , antiderivative size = 119, normalized size of antiderivative = 0.91 \[ \int \frac {\left (a+\frac {b}{x}\right )^n x}{c+d x} \, dx=\frac {\left (a+\frac {b}{x}\right )^n (b+a x) \left (a^2 c^2 \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )+(a c-b d) \left (a d (1+n) x+(-a c+b d n) \operatorname {Hypergeometric2F1}\left (1,1+n,2+n,1+\frac {b}{a x}\right )\right )\right )}{a^2 d^2 (a c-b d) (1+n) x} \]
((a + b/x)^n*(b + a*x)*(a^2*c^2*Hypergeometric2F1[1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)] + (a*c - b*d)*(a*d*(1 + n)*x + (-(a*c) + b*d*n)*Hyperg eometric2F1[1, 1 + n, 2 + n, 1 + b/(a*x)])))/(a^2*d^2*(a*c - b*d)*(1 + n)* x)
Time = 0.27 (sec) , antiderivative size = 140, normalized size of antiderivative = 1.07, number of steps used = 7, number of rules used = 6, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.333, Rules used = {1016, 899, 114, 174, 75, 78}
Below are the steps used by Rubi to obtain the solution. The rule number used for the transformation is given above next to the arrow. The rules definitions used are listed below.
\(\displaystyle \int \frac {x \left (a+\frac {b}{x}\right )^n}{c+d x} \, dx\) |
\(\Big \downarrow \) 1016 |
\(\displaystyle \int \frac {\left (a+\frac {b}{x}\right )^n}{\frac {c}{x}+d}dx\) |
\(\Big \downarrow \) 899 |
\(\displaystyle -\int \frac {\left (a+\frac {b}{x}\right )^n x^2}{\frac {c}{x}+d}d\frac {1}{x}\) |
\(\Big \downarrow \) 114 |
\(\displaystyle \frac {\int \frac {\left (a+\frac {b}{x}\right )^n \left (a c-\frac {b n c}{x}-b d n\right ) x}{\frac {c}{x}+d}d\frac {1}{x}}{a d}+\frac {x \left (a+\frac {b}{x}\right )^{n+1}}{a d}\) |
\(\Big \downarrow \) 174 |
\(\displaystyle \frac {\frac {(a c-b d n) \int \left (a+\frac {b}{x}\right )^n xd\frac {1}{x}}{d}-\frac {a c^2 \int \frac {\left (a+\frac {b}{x}\right )^n}{\frac {c}{x}+d}d\frac {1}{x}}{d}}{a d}+\frac {x \left (a+\frac {b}{x}\right )^{n+1}}{a d}\) |
\(\Big \downarrow \) 75 |
\(\displaystyle \frac {-\frac {a c^2 \int \frac {\left (a+\frac {b}{x}\right )^n}{\frac {c}{x}+d}d\frac {1}{x}}{d}-\frac {\left (a+\frac {b}{x}\right )^{n+1} (a c-b d n) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b}{a x}+1\right )}{a d (n+1)}}{a d}+\frac {x \left (a+\frac {b}{x}\right )^{n+1}}{a d}\) |
\(\Big \downarrow \) 78 |
\(\displaystyle \frac {\frac {a c^2 \left (a+\frac {b}{x}\right )^{n+1} \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {c \left (a+\frac {b}{x}\right )}{a c-b d}\right )}{d (n+1) (a c-b d)}-\frac {\left (a+\frac {b}{x}\right )^{n+1} (a c-b d n) \operatorname {Hypergeometric2F1}\left (1,n+1,n+2,\frac {b}{a x}+1\right )}{a d (n+1)}}{a d}+\frac {x \left (a+\frac {b}{x}\right )^{n+1}}{a d}\) |
((a + b/x)^(1 + n)*x)/(a*d) + ((a*c^2*(a + b/x)^(1 + n)*Hypergeometric2F1[ 1, 1 + n, 2 + n, (c*(a + b/x))/(a*c - b*d)])/(d*(a*c - b*d)*(1 + n)) - ((a *c - b*d*n)*(a + b/x)^(1 + n)*Hypergeometric2F1[1, 1 + n, 2 + n, 1 + b/(a* x)])/(a*d*(1 + n)))/(a*d)
3.3.86.3.1 Defintions of rubi rules used
Int[((b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[((c + d*x )^(n + 1)/(d*(n + 1)*(-d/(b*c))^m))*Hypergeometric2F1[-m, n + 1, n + 2, 1 + d*(x/c)], x] /; FreeQ[{b, c, d, m, n}, x] && !IntegerQ[n] && (IntegerQ[m] || GtQ[-d/(b*c), 0])
Int[((a_) + (b_.)*(x_))^(m_)*((c_) + (d_.)*(x_))^(n_), x_Symbol] :> Simp[(b *c - a*d)^n*((a + b*x)^(m + 1)/(b^(n + 1)*(m + 1)))*Hypergeometric2F1[-n, m + 1, m + 2, (-d)*((a + b*x)/(b*c - a*d))], x] /; FreeQ[{a, b, c, d, m}, x] && !IntegerQ[m] && IntegerQ[n]
Int[((a_.) + (b_.)*(x_))^(m_)*((c_.) + (d_.)*(x_))^(n_)*((e_.) + (f_.)*(x_) )^(p_), x_] :> Simp[b*(a + b*x)^(m + 1)*(c + d*x)^(n + 1)*((e + f*x)^(p + 1 )/((m + 1)*(b*c - a*d)*(b*e - a*f))), x] + Simp[1/((m + 1)*(b*c - a*d)*(b*e - a*f)) Int[(a + b*x)^(m + 1)*(c + d*x)^n*(e + f*x)^p*Simp[a*d*f*(m + 1) - b*(d*e*(m + n + 2) + c*f*(m + p + 2)) - b*d*f*(m + n + p + 3)*x, x], x], x] /; FreeQ[{a, b, c, d, e, f, n, p}, x] && ILtQ[m, -1] && (IntegerQ[n] || IntegersQ[2*n, 2*p] || ILtQ[m + n + p + 3, 0])
Int[(((e_.) + (f_.)*(x_))^(p_)*((g_.) + (h_.)*(x_)))/(((a_.) + (b_.)*(x_))* ((c_.) + (d_.)*(x_))), x_] :> Simp[(b*g - a*h)/(b*c - a*d) Int[(e + f*x)^ p/(a + b*x), x], x] - Simp[(d*g - c*h)/(b*c - a*d) Int[(e + f*x)^p/(c + d *x), x], x] /; FreeQ[{a, b, c, d, e, f, g, h}, x]
Int[((a_) + (b_.)*(x_)^(n_))^(p_.)*((c_) + (d_.)*(x_)^(n_))^(q_.), x_Symbol ] :> -Subst[Int[(a + b/x^n)^p*((c + d/x^n)^q/x^2), x], x, 1/x] /; FreeQ[{a, b, c, d, p, q}, x] && NeQ[b*c - a*d, 0] && ILtQ[n, 0]
Int[(x_)^(m_.)*((c_) + (d_.)*(x_)^(mn_.))^(q_.)*((a_) + (b_.)*(x_)^(n_.))^( p_.), x_Symbol] :> Int[x^(m - n*q)*(a + b*x^n)^p*(d + c*x^n)^q, x] /; FreeQ [{a, b, c, d, m, n, p}, x] && EqQ[mn, -n] && IntegerQ[q] && (PosQ[n] || !I ntegerQ[p])
\[\int \frac {\left (a +\frac {b}{x}\right )^{n} x}{d x +c}d x\]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n x}{c+d x} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n} x}{d x + c} \,d x } \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n x}{c+d x} \, dx=\int \frac {x \left (a + \frac {b}{x}\right )^{n}}{c + d x}\, dx \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n x}{c+d x} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n} x}{d x + c} \,d x } \]
\[ \int \frac {\left (a+\frac {b}{x}\right )^n x}{c+d x} \, dx=\int { \frac {{\left (a + \frac {b}{x}\right )}^{n} x}{d x + c} \,d x } \]
Timed out. \[ \int \frac {\left (a+\frac {b}{x}\right )^n x}{c+d x} \, dx=\int \frac {x\,{\left (a+\frac {b}{x}\right )}^n}{c+d\,x} \,d x \]